Invariants of rigid surface operators

Lusztig used the symbol invariant to describe the Springer correspondence for classical groups. Similarly, the fingerprint invariant can describe the Kazhdan-Lusztig map. Both invariants pertain to rigid semisimple operators labeled by pairs of partitions $(\lambda', \lambda'')$. It is conjectured that the symbol invariant is equivalent to the fingerprint invariant for rigid surface operators. In this study, we provide a proof of this conjecture. We classify the maps that preserve the fingerprint invariant and demonstrate that they also preserve the symbol invariant. Conversely, we classify the maps that preserve the symbol invariant and show that they also preserve the fingerprint invariant. The constructions of the symbol and fingerprint invariants in prior works are crucial to the proof. Additionally, we found that one condition in the definition of the fingerprint invariant is redundant for rigid surface operators. In the appendix, we present an alternative strategy to prove the equivalence of these invariants.